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Rsa

Dec 02, 2014

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Technology

 

  • 1. Public Key Cryptographyand the RSA Algorithm Cryptography and Network Security by William Stallings Lecture slides by Lawrie Brown Edited by Dick Steflik

2. Private-Key Cryptography

  • traditionalprivate/secret/single keycryptography usesonekey
  • Key is shared by both sender and receiver
  • if the key is disclosed communications are compromised
  • also known assymmetric , both parties are equal
    • hence does not protect sender from receiver forging a message & claiming is sent by sender

3. Public-Key Cryptography

  • probably most significant advance in the 3000 year history of cryptography
  • usestwokeys a public key and a private key
  • asymmetricsince parties arenotequal
  • uses clever application of number theory concepts to function
  • complementsrather thanreplaces private key cryptography

4. Public-Key Cryptography

  • public-key/two-key/asymmetriccryptography involves the use oftwokeys:
    • apublic-key , which may be known by anybody, and can be used toencrypt messages , andverify signatures
    • aprivate-key , known only to the recipient, used todecrypt messages , andsign(create)signatures
  • isasymmetricbecause
    • those who encrypt messages or verify signaturescannotdecrypt messages or create signatures

5. Public-Key Cryptography 6. Why Public-Key Cryptography?

  • developed to address two key issues:
    • key distribution how to have secure communications in general without having to trust a KDC with your key
    • digital signatures how to verify a message comes intact from the claimed sender
  • public invention due to Whitfield Diffie & Martin Hellman at Stanford U. in 1976
    • known earlier in classified community

7. Public-Key Characteristics

  • Public-Key algorithms rely on two keys with the characteristics that it is:
    • computationally infeasible to find decryption key knowing only algorithm & encryption key
    • computationally easy to en/decrypt messages when the relevant (en/decrypt) key is known
    • either of the two related keys can be used for encryption, with the other used for decryption (in some schemes)

8. Public-Key Cryptosystems 9. Public-Key Applications

  • can classify uses into 3 categories:
    • encryption/decryption(provide secrecy)
    • digital signatures(provide authentication)
    • key exchange(of session keys)
  • some algorithms are suitable for all uses, others are specific to one

10. Security of Public Key Schemes

  • like private key schemes brute forceexhaustive searchattack is always theoretically possible
  • but keys used are too large (>512bits)
  • security relies on alarge enoughdifference in difficulty betweeneasy(en/decrypt) andhard(cryptanalyse) problems
  • more generally thehardproblem is known, its just made too hard to do in practise
  • requires the use ofvery large numbers
  • hence isslowcompared to private key schemes

11. RSA

  • by Rivest, Shamir & Adlemanof MIT in 1977
  • best known & widely used public-key scheme
  • based on exponentiation in a finite (Galois) field over integers modulo a prime
    • nb. exponentiation takes O((log n) 3 ) operations (easy)
  • uses large integers (eg. 1024 bits)
  • security due to cost of factoring large numbers
    • nb. factorization takes O(elog n log log n ) operations (hard)

12. RSA Key Setup

  • each user generates a public/private key pair by:
  • selecting two large primes at random -p, q
  • computing their system modulusN=p.q
    • note(N)=(p-1)(q-1)
  • selecting at random the encryption keye
      • where 1< e